I should give a talk on something I'm working on, and I'd like to have a list, as complete as possible, of applications, in and out of functional analysis, of the following classical result by F. Riesz:

Riesz's lemma. Let $\mathcal V = (\mathbb V, \|\cdot\|)$ be a normed space over the normed field, $\mathcal K = (\mathbb K, |\cdot|)$, of real/complex numbers, $W$ a closed proper subspace of $\mathcal V$, and $\delta$ a real number with $0 < \delta < 1$. There then exists $x \in \mathcal V$ with $\|x\| = 1$ such that $\|x - y\| \ge \delta$ for all $y \in W$.

Classical applications of which I'm already aware:

  1. That the unit ball of a real/complex normed space $\mathcal V$ is compact iff $\mathcal V$ is finite-dimensional.
  2. The non-existence of certain measures for infinite-dimensional normed spaces.
  3. The spectral theorem for compact operators on a (complex) Banach space.

By the way, where can I find some focused discussion on the 2nd point of the above list? It seems to me that I read an excellent paper on the topic, some time ago, but I can't remember either the author(s), the journal, or other useful details, and I've started thinking that I dreamed of it.

Thanks in advance.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.